Analysis of Queuing Model for Machine Repairing System with Bernoulli Vacation Schedule

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1 Intenatonal Jounal of Mathematcs Tends and Technology Volume 10 Numbe 2 Jun 2014 Analyss of Queung Model fo Machne epang System wth Benoull Vacaton Schedule.K. Shvastava #1, Awadhesh Kuma Msha *2 1# Pofesso of Mathematcs, S.M.S. Govt. Model Scence College, Jwaj Unvesty, Gwalo, Inda 2 eseach Schola, S.M.S. Govt. Model Scence College, Jwaj Unvesty, Gwalo, Inda Abstact In ths pape, we consde a queueng model fo machne epang system wth Benoull vacaton schedule. The falue tmes, epa tmes and vacaton tmes ae all assumed to be exponentally dstbuted. In congeston, the seve may ncease the epa ate wth pessue coeffcent to educe the queue length. We assume that the seve begns the wokng vacaton when the system s empty. The seve may go fo a vacaton of andom length wth pobablty p o may contnue to epa the next (f avalable) faled machne wth pobablty q = 1 p. The whole system s modelled as a fnte state Makov Chan and ts steady state dstbuton s obtaned by matx ecusve appoach. Keywods Machne epa, Benoull vacaton, pessue coeffcent, Makov Chan, Matx-ecusve method. I. INTODUCTION The machnng system have pevaded evey feld of ou lves n dffeent actvtes ensung ou almost total dependence on machnes. As the tme passes a machne become to pone falue. The falue of machnes may esult n loss of poducton, money, goodwll etc. If at any tme a machne fals, t s sent to the epa faclty fo epa. In ths system we consde a goup of M opeatng machnes and epamen (seves) n the epa faclty. To avod any loss of poducton the plant o company always keeps some standby machnes so that a standby machne can mmedately act as a substtute when an opeatng machne fals. To mnmze the machne falue duaton, we consde a stuaton whee the epa ate s nceasng when thee ae faled machnes watng fo epa. The featue s modeled by a "sevce pessue coeffcent ". Ths coeffcent s a postve constant and ndcates the degee to whch the seves ncease the epa ate n ode to educe the numbe of faled machnes mmedately. In moden age, the falue and epas ae coupled events n a typcal machnng system. The mpotant contbuton n the aeas of machne epa poblem ae due to Goss and Has [4], Gupta, S.M. [5], Jan, M. [7] and Jan, M. [8]. An unelable etal queue wth two phases of sevce and Benoull admsson mechansms was studed by Choudhay, G. and Daka, K. [2]. A two stage batch aval queueng system wth a modfed Benoull schedule vacaton unde N-polcy was analysed by Choudhay, G. [3]. Goss and Has [4] studed fundamentals of queueng theoy. Gupta, S.M. [5] developed machne ntefeence poblem wth wam spaces seve vacatons and exhaustve sevce. Jan, M. [9] povded tansent analyss of machne epang system wth sevce nteupton mxed standby and poty. Jan, M., Shama,G.C. and Sngh, M.[10] dscussed dffuson pocess fo mult-epaman machnng system wth spaes and balkng. Jan, M., Sulekha an [11] ntoduced avalablty analyss of epaable system wth wam standby swtchng falue and eboot delay. Jan, M., Chanda Shekha and Shaln Shukla [12] dscussed queueng analyss of a mult-component machnng system havng unelable heteogeneous seves and mpatent customes. Ke, J.C., Hsu, Y.L., Lu, T.H. and Zhang, Z.G. [13] analyzed computatonal analyss of machne epa poblem wth unelable mult-epamen. Ke, J.C. Lee, S.I. and Lou, C.H. [14] developed machne epa poblem n poducton systems wth spaes and seve vacatons. Ke J.C. Wu, C.H. and Pean, W.L. [15] pefomed algothm analyss of the mult-seve system wth modfed Benoull schedule. Ke, J.C., Wu,C.H. and Pean, W.L. [16] gave analyss of an nfnte mult-seve queue wth an optonal sevce. In machnng systems, queueng systems wth vacatons have many applcatons wokng n ndustal such as manufactung and poducton systems. When thee s no faled unt-pesent n the system, what should a epaman do? Fo ths, nstead of emanng dle dung ths peod, the epamen may go fo a vacaton and can utlze ths tme to do some othe wok such as peventve mantenance, pope aangement of tools etc. ove last fou decades, a substantal amount of wok has been done to examne queung systems wth vacatons. Kelson J., and Sev, L.D. [17] analyzed oscllatng andom walk models fo GI/G/1 vacaton systems wth Benoull schedule. Khoam, E. [18] developed an optmal model by dynamc numbes of epaman nfnte populaton queueng system. On the mult-seve machne ntefeence wth modfed Benoull vacatons was studed by Lu, Hsn, T. and Ke J.C. [19]. A two seve queue wth Benoull schedule and sngle vacaton polcy was developed by Madan, K.C.,W. Abu Dayyah and Tayyan, F. [20]. Maheshwa Supya, Al Saza [21] studed machne epa poblem ISSN: Page 85

2 Intenatonal Jounal of Mathematcs Tends and Technology Volume 10 Numbe 2 Jun 2014 wth mxed spaes, balkng and enegng. Maheshwa Supya et al. [22] developed Machne epa poblem wth K-type wam spaes multple vacatons and enegng. Wang, K.H. and Wu, J.D. [23] developed cost analyss of the M/M/ machne epa poblem wth spaes and two modes of falue. Poft analyss of the M/ M/ // machne epa poblem wth balkng, enegng and standby swtchng falue was gven by Wang, K.H., Ke, J. B. and Ke, J.C., [24]. Wang, K.H. Chen, W.L. and Yang, D.Y. [25] developed optmal management of the machne epa poblem wth wokng vacaton, Newton's method. Analyss of mult-seve queue wth a sngle vacaton (e, d) polcy was studed by Xu, X and Zhang, Z.G., [26]. Modelng of mult-seve epa poblem wth swtchng falue and eboot delay unde elated poft analyss was developed by Yng, Ln, Hsu, Leu J.C.and Tsu-Hsn, Lu and C.H. Wu. [27]. Yue, D. Yue, W., and Q, H. [28] vsualzed pefomance analyss and optmzaton of a machne epa poblem wth wam spaes and two heteogeneous epamen. Fgue-1. Machne epamen poblem. II. MODEL DESCIPTION Consde L = M + S homogeneous machnes, whee M machnes ae opeatng S machnes ae avalable as standbys. Thee ae epamen, The epa tmes ae assumed to follow ndependent and dentcally exponental dstbuton wth ate. Each of the opeatng machnes fals accodng to a Posson pocess wth paamete (o whee 0 ). When an actve (o standby) machne fals, t s mmedately eplaced by any avalable standby and s epaed n the ode of beakdowns by any avalable seve. At each epa completon nstant of a seve, the seve nspects the system state and decdes whethe to leave fo a vacaton of andom length wth pobablty p o contnue to epa the next faled machne f any wth pobablty q = 1 p. When the system s empty, the seve begns a wokng vacaton, and the vacaton duaton follows an exponental dstbuton wth mean duaton 1/. When a wokng vacaton temnates and the system s empty, the seve stats anothe wokng vacaton. epa tmes dung a vacaton peod ae accodng to exponental dstbuton wth mean 1/. epa ate s defned as follows: n ; n 0,1, 2,..., 1 n n( 1) ; whee, 0,. ( 1) n L n whee s a pessue coeffcent epesentng the degee to whch the epa ate s effected by the numbe of faled machnes n the system. We consde a stuaton n whch seves ae unde wok pessue. When numeous faled machnes ae watng fo epa sevces and few seves ae avalable, the seves may pefom bette unde easonable pessue. In ths pape we have studed model pesented by Lu, Hsn, T. and Ke,J.C. [19]. III. 3. THE GOVENING EQUATIONS Consde a mult-seve machne ntefeence poblem wth modfed Benoull vacaton unde a sngle vacaton polcy. In steady state, we have followng pobabltes. P (n) : Pob. that thee ae n faled machnes n the system when thee ae vacatonng seves, n = 0, 1,..,L and = 0, 1, 2,..,. The falue ate n and epa ate M ( S n) ; 0 n S n ( L n) ; S 1 n L n ae defned as follows: ISSN: Page 86

3 and Intenatonal Jounal of Mathematcs Tends and Technology Volume 10 Numbe 2 Jun 2014 n n( 1) ( )( 1) n n; n 1, ( ) ; n, 1, The state tanston ate dagam fo a mult-seve machne epa system wth modfed Benoull vacaton schedule s gven below: Fgue-2. State-tanston ate dagam fo a mult-seve machne epa system wth modfed Benoull vacaton schedule. Accodng to fgue-2 state tanston ate dagam whee (m, n) of ccle denotes the state that thee ae m vacaton seves ae n faled machnes n the system, steady state equatons of the machne epa model ae gven as follows: () When 0 P (0) q P (1) P (0) (1) (0) ( ) P ( n) P ( n 1) P ( n) q P ( n 1); 1 n 1 (2) 0 (0) n n 0 n1 0 1 n1 0 ( ) P ( ) P ( 1) P ( ) P ( 1); (3) 0 (0) ( ) P ( n) P ( n 1) P ( n) P ( n 1); 1 n L 1 (4) 0 (0) n n 0 n1 0 1 n1 0 P ( L) P ( L 1) P ( L) (5) 0 L 0 L1 0 1 () When, 1 K 1 ( ) P (0) q P (1) ( 1) P (0) p P (0) (6) ISSN: Page 87

4 Intenatonal Jounal of Mathematcs Tends and Technology Volume 10 Numbe 2 Jun 2014 ( ) P ( n) P ( n 1) ( 1) P ( n) q P ( n 1) n n n1 1 n1 1 p n1p 1( n 1); 1 n 1 (7) ( ) P ( ) P ( 1) ( 1) P ( ) 1 1 P ( 1) p P ( 1); (8) ( n n ) P ( n) n1p ( n 1) ( 1) P 1( n) n1p ( n 1) 1 n L 1 (9) ( ) P ( L) P ( L 1) ( 1) P ( L) (10) () When, L L1 1 ( ) P (0) p P (1) (11) ( ) P ( n) P ( n 1); 1 n L 1 (12) n n1 P ( L) P ( L 1) (13) L1 Snce the closed fom pobablty solutons of equatons (1)-(13) ae too complcate to develop explct expessons by usng a ecusve method. Hence we use matx-geometc methods to analyss ths poblem we fnd that the equatons (1)-(13) of the pesent model can be expessed n the matx fom. To solve the flow balance equatons (1)-(13) fo the statonay dstbuton, we constuct the tanston ate matx Q whch s as follows: D0 U L D U 0 L1 D2 U 2 Q L2 D3 U3 0 L 2 D 1 U 0 0 L 1 D Matces D 0 and D 1 flte those pats of the Makov pocess whch coespond to no-aval and aval tanstons espectvely. Sub-matces of matx Q ae gven as follows: p U p v0, , 1, u v 0 u2, 2, v D u L1, vl, L ul, vl, ISSN: Page 88

5 Intenatonal Jounal of Mathematcs Tends and Technology Volume 10 Numbe 2 Jun 2014 ( 0 ) ; 1 1 and j 0 ( 0 u j ); 1 1 and 2 j L 1 v j, ( L ) ; 1 1 and j L ( j ) ; and 0 j L 1 ; and j L and q j ; 1 1 and 1 j 1 v j, j ; 1 1 and j L 0 ; othewse Let be pattoned confomally wth Q,.e. [,,... ] (14) 0 1 Whee, s a 1 (L + 1) ow vecto whch s gven below: [ P (0), P (1),... P ( L)]; 0,1,..., Now Q = 0 (15) (16) 0D0 1L0 0 1U D 1L 0; 1 1 (17) and (18) 1U D 0 Afte some outne substtutons, we have 1 1U D (19) 1 X; 1 1 (20) ( D X L ) 0 (21) Whee, And, X U D X L 1 ( 1 ) ; 1 2 X U ( D U D L ) k 1 Equaton (21) detemnes 0 and equatons (19) and (20) detemned detemned by the nomalzng equaton 1,..., 1 upto a constant. Ths constant can be 0 e 1 (22) Whee, e s a column vecto wth all elements equal to one.e. e (1,1,...,1). ISSN: Page 89

6 Intenatonal Jounal of Mathematcs Tends and Technology Volume 10 Numbe 2 Jun 2014 IV. SYSTEM PEFOMANCE MEASUES AND COST ANALYSIS Vaous pefomance measues of machnng system n tems of pobabltes can be obtaned. We defne assumptons and notatons whch ae computed s follows: The aveage no. of faled machnes n the system L E( f ) np ( n ) 0 n0 The aveage no. of vacatonng seves n the system E( v) P ( n ) 0 n0 The aveage no. of dle seves n the system 1 E( ) ( ) P ( n ) 0 n0 The aveage no. of busy seves n the system E( b) E( w) E( ) The aveage no. of opeatng machnes n the system L E( O) M ( n S) P ( n ) 0 ns 1 The aveage no. of standby machnes n the system S E( s) ( S n) P ( n ) 0 n0 Benson and Cox [1] defned the machne avalablty and the seve utlzaton as follows: Machne avalablty (M.A.) = L E f L ( ) (23) Seve utlzaton (the facton of busy seves) O.U. = E( b). (24) Now we detemne the optmal amount of esouces to mantan the system avalablty at a cetan level. Defne, the steady state pobablty that at least M machnes ae n opeaton (system avalablty) = Av; and the mnmum facton of Av s gven by A 0 : The poducton system eques a mnmum of M machnes n opeaton. The cost pe unt tme of each machne downtme occus when thee ae less than M opeatng machnes n the system. We select the followng cost elements. c(h) = Cost pe unt tme fo a faled machne. c(e) = Cost pe unt tme of a faled machne afte all standbys ae exhausted. ISSN: Page 90

7 Intenatonal Jounal of Mathematcs Tends and Technology Volume 10 Numbe 2 Jun 2014 c(w) = Cost pe unt tme fo a machne functonng as standby. c(b) = cost pe unt tme fo a busy seve. c() = Cost pe unt tme fo an dle seve. c(v) = Cost pe unt tme fo a vacatonng seve. c(s) = Cost pe unt tme fo offeng sevce ate fo a faled machne. c(o) = Cost of (loss) opeatve utlzaton. Wth these costs, we consde, s, as decson vaables and wte the total cost functon as follows: Cost functon F(, S, ) = c(h) E(f) c(e) [ M E(O)] + c(w) E(s) + c(b) E(b) + c() E() c(v) E(v) + c(s) + c(o) (1 O.U.). (26) Subject to Av A 0 (27) Thee s a tade-off between the cost and the avalablty of the system. The cost functon (26) s hghly nonlnea and complex. Ou am s to desgnate optmal values of some contollable paametes fo ths system unde the cost functon gven n equaton (26). Two dscete vaables, S and one contnuous vaable ae consdeed. Ou man objectve s to detemne the optmal values of, S and so as to mnmze ths cost functon F(, S, ) subject to equaton (27). V. CONCLUSIONS In pesent pape, we have studed a machne epang system wth Benoull vacaton schedule. In ths atcle we examne the effect of epa pessue on the system pefomance, such a system can be fequently used to model as a system wth an electonc equpment. We have developed a Makov Chan model and obtaned the statonay dstbuton usng matx ecusve appoach also we have developed the expected cost functon. EFEENCES [1] [1] Benson. F., and Cox,D..(1951): The poducton of machne epang attenton and andom ntevals. Jounal of the oyal Statstcal Socety,B, 13, pp [2] [2] Choudhuy, G. and Daka, K. (2009): An MX / G / 1 unelable etal queue wth two phases of sevce and Benoull admsson mechansm. Appled Mathematcs and Computaton, 215, pp [3] [3] Choudhuy,G.(2005): A two stage batch aval queueng system wth a modfed Benoull schedule vacaton unde N-polcy. Mathematcal and Compute Modelng, 42, pp [4] [4] Goss, D. and Has, C.M. (1985): Fundamentals of queueng theoy, 2nd edton John Wley and Sons, New Yok. [5] [5] Gupta, S.M. (1997): Machne ntefeence poblem wth wam spaes, seve vacatons and exhaustve sevce. Pefomance Evaluaton, 29(3), pp [6] [6] Hses, Y. C and Wang, K.H. (1995): elablty of a epaable system wth spaes and emovable epamen. Mcoelectoncs and elablty, 35, pp [7] [7] Jan, M. (1998): M/M// machne epa poblem wth spaes and addtonal seves. Indan Jounal of Pue and Appled Mathematcs, Vol. 29, no. 5, pp [8] [8] Jan, M. (2003): N-polcy edundant epaable system wth addtonal epaman. OPSEACH, 40(2), pp [9] [9] Jan, M. (2013): Tansent Analyss of machnng system wth sevce nteupton, mxed standbys and poty. Intenatonal Jounal of Mathematcs n Opeatons eseach, Vol. 5, No.5, pp (Indescence). [10] [10] Jan, M. Shama, G.C. and Sngh, M. (2002): Dffuson pocess fo mult-epaman machnng system wth spaes and balkng. Intenatonal Jounal of Engneeng Scence 15(1), pp [11] [11] Jan, M., and Sulekha, an (2013): Avalablty analyss of epaable system wth wam standby, swtchng falue and eboot delay. Intenatonal jounal of Mathematcs n opeatons eseach Vol.5, No.1, pp , (Indescence). [12] [12] Jan, M., Chanda Shekha and Shukla, Shaln (2012): Queueng analyss of a mult-component machnng system havng unelable heteogeneous seves and mpatent customes, vol. 2(3), pp [13] [13] Ke, J.C., Hsu, Y.L., Lu, T.H., and Zhang Z.G. (2013): Computatonal analyss of machne epa poblem wth unelable mult-epamen. Computes and Opeatons eseach, 40(3), pp [14] [14] Ke, J.C., Lee, S.I. and Lou, C.H. (2009): Machne epa poblem n poducton systems wth spaes and seve vacatons. AIO, Opeatons eseach Vol. 43, No. pp [15] [15] Ke, J.C., Wu., C.H. and Pean, W.L.(2011) : Algothm analyss of the mult-seve system wth modfed Benoull schedule. Appled Mathematcal Modellng. 35, pp [16] [16] Ke,J.C., Cha-Huang,Wu., and Pean, W.L.(2013): Analyss of an nfnte mult-seve queue wth an optonal sevce. Computes and Industal Engneeng. 62(2), pp ISSN: Page 91

8 Intenatonal Jounal of Mathematcs Tends and Technology Volume 10 Numbe 2 Jun 2014 [17] [17] Kelson, J., and Sev, L.D.(1986): Oscllatng andom walk models fo GI/G/1 Vacaton systems wth Benoull Schedule. J. Appl. Pob. Vol. 23, pp [18] [18] Khoam, E. (2008): An Optmal queung model by dynamc numbes of epaman n fnte populaton queueng system. Qualty Technology and Quanttatve Management, Vol.5, No. 4, pp [19] [19] Lu, Hsn, T., and Ke, J.C. (2014): On the mult-seve machne ntefeence wth modfed Benoull vacatons. Jounal of Industal and Management Optmzaton. vol.10, no.4, pp [20] [20] Madan, K.C. W. Abu-dayyeh and Tayyan, F. (2003): A two seve queue wth Benoull schedules and a sngle vacaton polcy. Appled Mathematcs and Computaton.145, pp [21] [21] Maheshwa, Supya and Al, Shaza. (2013): Machne epa Poblem wth Mxed Spaes Balkng and enegng. Intenatonal Jounal of Theoetc and Appled Scences, 5(1),pp [22] [22] Maheshwa, Supya, et al. (2010), Machne epa poblem wth K-type wam spaes, multple vacatons fo epaman and enegng. Intenatonal Jounal of Engneeng and Technology, Vol. 2(4), pp [23] [23] Wang K.H. and Wu, J.D. (1995): Cost analyss of the M/M/ machne epa poblem wth spaes and two modes of falue. Jounal of the Opeatonal eseach Socety, 46, [24] [24] Wang, K.H. Ke, J.B. and Ke. J.C. (2007), Poft analyss of the M/M/ machne epa poblem wth balkng, enegng and standby swtchng falue, Computes and Opeatons eseach 34(3), pp [25] [25] Wang, K.H., Chen, W.L. and Yang, D.Y. (2009): Optmal management of the machne epa poblem wth wokng vacaton, Newton s method. Jounal of Computatonal and Appled Mathematcs. 233, pp [26] [26] Xu.X. and Zhang, Z.G. (2006): Analyss of mult-seve queue wth a sngle vacaton (e,d)-polcy. Pefomance evaluaton, vol. 63, pp [27] [27] Yng- Ln,Hsu., Ke,J.C.,Tzu -Hsn Lu, and Cha-huang, Wu.(2014): Modelng of mult-seve epa poblem wth swtchng falue and eboot delay unde elated poft analyss. Computes & Industal Engneeng,69,pp [28] [28] Yue, D., Yue, W., and Q. H. (2013): Pefomance Analyss and Optmzaton of a Machne epa Poblem wth wam spaes and two heteogeneous epamen. Optmzaton and Engneeng, Vol. 3, ssue 4, pp ISSN: Page 92